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FuXi-S2S: A machine learning model that outperforms conventional global subseasonal forecast models

Published 15 Dec 2023 in physics.ao-ph, cs.AI, and cs.LG | (2312.09926v2)

Abstract: Skillful subseasonal forecasts are crucial for various sectors of society but pose a grand scientific challenge. Recently, machine learning based weather forecasting models outperform the most successful numerical weather predictions generated by the European Centre for Medium-Range Weather Forecasts (ECMWF), but have not yet surpassed conventional models at subseasonal timescales. This paper introduces FuXi Subseasonal-to-Seasonal (FuXi-S2S), a machine learning model that provides global daily mean forecasts up to 42 days, encompassing five upper-air atmospheric variables at 13 pressure levels and 11 surface variables. FuXi-S2S, trained on 72 years of daily statistics from ECMWF ERA5 reanalysis data, outperforms the ECMWF's state-of-the-art Subseasonal-to-Seasonal model in ensemble mean and ensemble forecasts for total precipitation and outgoing longwave radiation, notably enhancing global precipitation forecast. The improved performance of FuXi-S2S can be primarily attributed to its superior capability to capture forecast uncertainty and accurately predict the Madden-Julian Oscillation (MJO), extending the skillful MJO prediction from 30 days to 36 days. Moreover, FuXi-S2S not only captures realistic teleconnections associated with the MJO, but also emerges as a valuable tool for discovering precursor signals, offering researchers insights and potentially establishing a new paradigm in Earth system science research.

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References (69)
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Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pegion, K., et al.: The subseasonal experiment (subx): A multimodel subseasonal prediction experiment. Bulletin of the American Meteorological Society 100(10), 2043–2060 (2019) (4) White, C.J., et al.: Advances in the application and utility of subseasonal-to-seasonal predictions. Bulletin of the American Meteorological Society 103(6), 1448–1472 (2022) (5) Domeisen, D.I., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (6) Merryfield, W.J., et al.: Current and emerging developments in subseasonal to decadal prediction. Bulletin of the American Meteorological Society 101(6), 869–896 (2020) (7) Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 White, C.J., et al.: Advances in the application and utility of subseasonal-to-seasonal predictions. Bulletin of the American Meteorological Society 103(6), 1448–1472 (2022) (5) Domeisen, D.I., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (6) Merryfield, W.J., et al.: Current and emerging developments in subseasonal to decadal prediction. Bulletin of the American Meteorological Society 101(6), 869–896 (2020) (7) Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (6) Merryfield, W.J., et al.: Current and emerging developments in subseasonal to decadal prediction. Bulletin of the American Meteorological Society 101(6), 869–896 (2020) (7) Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Merryfield, W.J., et al.: Current and emerging developments in subseasonal to decadal prediction. Bulletin of the American Meteorological Society 101(6), 869–896 (2020) (7) Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. 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Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (6) Merryfield, W.J., et al.: Current and emerging developments in subseasonal to decadal prediction. Bulletin of the American Meteorological Society 101(6), 869–896 (2020) (7) Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Merryfield, W.J., et al.: Current and emerging developments in subseasonal to decadal prediction. Bulletin of the American Meteorological Society 101(6), 869–896 (2020) (7) Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (6) Merryfield, W.J., et al.: Current and emerging developments in subseasonal to decadal prediction. Bulletin of the American Meteorological Society 101(6), 869–896 (2020) (7) Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Merryfield, W.J., et al.: Current and emerging developments in subseasonal to decadal prediction. Bulletin of the American Meteorological Society 101(6), 869–896 (2020) (7) Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  4. Domeisen, D.I., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (6) Merryfield, W.J., et al.: Current and emerging developments in subseasonal to decadal prediction. Bulletin of the American Meteorological Society 101(6), 869–896 (2020) (7) Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Merryfield, W.J., et al.: Current and emerging developments in subseasonal to decadal prediction. Bulletin of the American Meteorological Society 101(6), 869–896 (2020) (7) Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Mouatadid, S., et al.: Adaptive bias correction for improved subseasonal forecasting. Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. 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Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. 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(2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. 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Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. 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Nature Communications 14(1), 3482 (2023) (8) Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. 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Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., Robertson, A.W., Anderson, D.L.: Subseasonal to seasonal prediction project: Bridging the gap between weather and climate. Bulletin of the World Meteorological Organization 61(2), 23 (2012) (9) Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. 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Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Sub-seasonal predictions. ECMWF Tech. Memo. 738 (2014) (10) Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. 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Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmos. Sci. 20(2), 130–148 (1963) (11) Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weyn, J.A., Durran, D.R., Caruana, R., Cresswell-Clay, N.: Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models. Journal of Advances in Modeling Earth Systems 13(7), 2021–002502 (2021) (12) Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. 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Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. 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(2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. 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Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. 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Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F.: Evolution of ecmwf sub-seasonal forecast skill scores. Quarterly Journal of the Royal Meteorological Society 140(683), 1889–1899 (2014) (13) Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Saha, S., et al.: The ncep climate forecast system version 2. Journal of climate 27(6), 2185–2208 (2014) (14) Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Monhart, S., et al.: Skill of subseasonal forecasts in europe: Effect of bias correction and downscaling using surface observations. Journal of Geophysical Research: Atmospheres 123(15), 7999–8016 (2018) (15) Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Domeisen, D.I., White, C.J., Afargan-Gerstman, H., Muñoz, Á.G., Janiga, M.A., Vitart, F., Wulff, C.O., Antoine, S., Ardilouze, C., Batté, L., et al.: Advances in the subseasonal prediction of extreme events: relevant case studies across the globe. Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. 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Bulletin of the American Meteorological Society 103(6), 1473–1501 (2022) (16) Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. 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Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nowak, K., Webb, R., Cifelli, R., Brekke, L.: Sub-seasonal climate forecast rodeo. In: 2017 AGU Fall Meeting, New Orleans, LA, pp. 11–15 (2017) (17) Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: Outcomes of the wmo prize challenge to improve subseasonal to seasonal predictions using artificial intelligence. Bulletin of the American Meteorological Society 103(12), 2878–2886 (2022) (18) Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Cohen, J., et al.: S2s reboot: An argument for greater inclusion of machine learning in subseasonal to seasonal forecasts. Wiley Interdisciplinary Reviews: Climate Change 10(2), 00567 (2019) (19) Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. 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Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Richardson, D.S.: Measures of skill and value of ensemble prediction systems, their interrelationship and the effect of ensemble size. Quarterly Journal of the Royal Meteorological Society 127(577), 2473–2489 (2001) (20) Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Pathak, J., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. Preprint at https://arxiv.org/abs/2202.11214 (2022) (21) Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lam, R., et al.: Learning skillful medium-range global weather forecasting. Science (2023) (22) Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. 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Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. 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(2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. 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Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. Nature (2023) (23) Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Bi, K., et al.: Accurate medium-range global weather forecasting with 3d neural networks. 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In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, L., et al.: Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 1–11 (2023) (24) Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. 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Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. 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Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhong, X., Chen, L., , Liu, J., Liu, C., Qi, Y., Li, H.: FuXi-Extreme: Improving extreme rainfall and wind forecasts with diffusion model (2023) (25) Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Haiden, T., et al.: Evaluation of ECMWF forecasts, including the 2021 upgrade (2021) (26) Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. 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Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hersbach, H., et al.: The era5 global reanalysis. Q. J. R. Meteorol. Soc. 146(730), 1999–2049 (2020) (27) He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 He, S., Li, X., DelSole, T., Ravikumar, P., Banerjee, A.: Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances. Proceedings of the AAAI Conference on Artificial Intelligence 35(1), 169–177 (2021) (28) Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. 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Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. 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Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Friedman, J.H.: Greedy function approximation: a gradient boosting machine. Annals of statistics, 1189–1232 (2001) (29) Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, pp. 785–794 (2016) (30) Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology 58(1), 267–288 (1996) (31) Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. 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Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. 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Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. 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(2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. 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Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jalali, A., Ravikumar, P., Sanghavi, S.: A dirty model for multiple sparse regression. IEEE Transactions on Information Theory 59(12), 7947–7968 (2013) (32) Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in neural information processing systems 27, 3104–3112 (2014) (33) Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. 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Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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(2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kiefer, S.M., Lerch, S., Ludwig, P., Pinto, J.G.: Can machine learning models be a suitable tool for predicting central european cold winter weather on subseasonal to seasonal time scales? Artificial Intelligence for the Earth Systems 2(4), 1–16 (2023) (34) Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  33. Breiman, L.: Random forests. Machine learning 45, 5–32 (2001) (35) de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  34. de Andrade, F.M., Coelho, C.A., Cavalcanti, I.F.: Global precipitation hindcast quality assessment of the subseasonal to seasonal (s2s) prediction project models. Climate Dynamics 52(9-10), 5451–5475 (2019) (36) Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ecmwf ensemble prediction system: Methodology and validation. Quarterly Journal of the Royal Meteorological Society 122(529), 73–119 (1996) (37) Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  36. Madden, R.A., Julian, P.R.: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. Journal of Atmospheric Sciences 28(5), 702–708 (1971) (38) Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Madden, R.A., Julian, P.R.: Description of global-scale circulation cells in the tropics with a 40–50 day period. Journal of Atmospheric Sciences 29(6), 1109–1123 (1972) (39) Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation. Reviews of Geophysics 43(2) (2005) (40) Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  39. Zhang, C.: Madden-julian oscillation: Bridging weather and climate. Bulletin of the American Meteorological Society 94(12), 1849–1870 (2013) (41) Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhang, C., et al.: Cracking the mjo nut. Geophysical Research Letters 40(6), 1223–1230 (2013) (42) Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. 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Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  41. Neena, J., et al.: Predictability of the madden–julian oscillation in the intraseasonal variability hindcast experiment (isvhe). Journal of Climate 27(12), 4531–4543 (2014) (43) Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  42. Kim, H., Vitart, F., Waliser, D.E.: Prediction of the madden–julian oscillation: A review. Journal of Climate 31(23), 9425–9443 (2018) (44) Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. 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Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  43. Jiang, X., et al.: Fifty years of research on the madden-julian oscillation: Recent progress, challenges, and perspectives. Journal of Geophysical Research: Atmospheres 125(17), 2019–030911 (2020) (45) Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., et al.: Improving the prediction of the madden–julian oscillation of the ecmwf model by post-processing. Earth System Dynamics 13(3), 1157–1165 (2022) (46) Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. 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Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  45. Kim, H., Ham, Y.G., Joo, Y.S., Son, S.W.: Deep learning for bias correction of mjo prediction. Nature Communications 12(1) (47) Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Silini, R., Barreiro, M., Masoller, C.: Machine learning prediction of the madden-julian oscillation. npj Climate and Atmospheric Science 4(1), 57 (2021) (48) Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. 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Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  47. Toms, B.A., Kashinath, K., Yang, D., et al.: Testing the reliability of interpretable neural networks in geoscience using the madden–julian oscillation. Geoscientific Model Development 14(7), 4495–4508 (2021) (49) Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. 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(2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Delaunay, A., Christensen, H.M.: Interpretable deep learning for probabilistic mjo prediction. Geophysical Research Letters 49(16), 2022–098566 (2022) (50) Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wheeler, M.C., Hendon, H.H.: An all-season real-time multivariate mjo index: Development of an index for monitoring and prediction. Monthly weather review 132(8), 1917–1932 (2004) (51) Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. 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(2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Lim, Y., Son, S.-W., Kim, D.: Mjo prediction skill of the subseasonal-to-seasonal prediction models. Journal of Climate 31(10), 4075–4094 (2018) (52) Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. 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Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Zhu, B., Wang, B.: The 30-60-day convection seesaw between the tropical indian and western pacific oceans. Journal of the Atmospheric Sciences 50, 184–199 (1993) (53) Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Walker, G.T.: Correlations in seasonal variations of weather. viii, a further study of world weather. Men. Indian Meteor. Dept. 24, 275–332 (1924) (54) Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R.H.: Influence of the heat source anomaly over the western tropical pacific for the subtropical high over east asia. In: International Conference on the General Circulation of East Asia. Chendu, China, April 10-15, 1987, pp. 40–50 (1987) (55) Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Huang, R., Li, W.: Influence of heat source anomaly over the western tropical pacific on the subtropical high over east asia and its physical mechanism. Chin. J. Atmos. Sci 12(s1), 107–116 (1988) (56) Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Nitta, T.: Convective activities in the tropical western pacific and their impact on the northern hemisphere summer circulation. Journal of the Meteorological Society of Japan. Ser. II 65(3), 373–390 (1987) (57) Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Epstein, E.S.: A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology (1962-1982) 8(6), 985–987 (1969) (58) Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wilks, D.S.: Statistical Methods in the Atmospheric Sciences vol. 100, 3rd edn. (2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. 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Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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(2011) (59) Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Hong, C.-C., Huang, A.-Y., Hsu, H.-H., Tseng, W.-L., Lu, M.-M., Chang, C.-C.: Causes of 2022 pakistan flooding and its linkage with china and europe heatwaves. npj Climate and Atmospheric Science 6(1), 163 (2023) (60) Dunstone, N., Smith, D.M., Hardiman, S.C., Davies, P., Ineson, S., Jain, S., Kent, C., Martin, G., Scaife, A.A.: Windows of opportunity for predicting seasonal climate extremes highlighted by the pakistan floods of 2022. Nature Communications 14(1), 6544 (2023) (61) Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Vitart, F., et al.: The subseasonal to seasonal (s2s) prediction project database. Bulletin of the American Meteorological Society 98(1), 163–173 (2017) (62) Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. 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Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. 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Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Buizza, R.: Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. 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Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. 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Quarterly Journal of the Royal Meteorological Society 145(S1), 1–11 (2019) (63) Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. 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  62. Adler, R.F., Sapiano, M.R., Huffman, G.J., Wang, J.-J., Gu, G., Bolvin, D., Chiu, L., Schneider, U., Becker, A., Nelkin, E., et al.: The global precipitation climatology project (gpcp) monthly analysis (new version 2.3) and a review of 2017 global precipitation. Atmosphere 9(4), 138 (2018) (64) Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Stan, C., et al.: Advances in the prediction of mjo teleconnections in the s2s forecast systems. Bulletin of the American Meteorological Society 103(6), 1426–1447 (2022) (65) Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. Preprint at https://arxiv.org/abs/1412.6980 (2017) (66) Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. 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Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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  65. Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2017) (67) Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  66. Weigel, A.P., Liniger, M.A., Appenzeller, C.: The discrete brier and ranked probability skill scores. Monthly Weather Review 135(1), 118–124 (2007) (68) Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  67. Brier, G.W.: Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review 78(1), 1 (1950) (69) Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  68. Rashid, H.A., Hendon, H.H., Wheeler, M.C., Alves, O.: Prediction of the madden–julian oscillation with the poama dynamical prediction system. Climate Dynamics 36, 649–661 (2011) (70) Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377 Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
  69. Wang, S.: A Multivariate Index for Tropical intraseasonal Oscillations based on the Seasonally-varying Modal Structures. Zenodo (2021). https://doi.org/10.5281/zenodo.5806377. https://doi.org/10.5281/zenodo.5806377
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